Center of Mass Gravity

Textbooks always state that the pull of earth's gravity on an object above its surface can be calculated by Newton's law assuming the mass of the earth is concentrated to a point at its centre. As this is not obvious, I decided to prove it to myself by calculus, summing the contribution from each point within the earth, but got a different and more complex answer. Can you comment?
- Peter Fry (age 81)
England

So long as you assume that the Earth is spherically symmetric, you should get Newton's answer. He obtained it the same way you tried, by integral calculus, with the extra step that he had to invent integral calculus to solve this problem. You can see immediately by symmetry that the direction must point toward the center. Calculus is used to show that the strength, for all points outside the Earth's mass,  is the same as if all the mass were at the center.

Mike W.

(published on 03/30/2018)